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Applications of Exponential Maps to Epimorphism and Cancellation Problems
No full textThroughout this talk k will denote a field. The talk is primarily divided into two parts. In the first part we will discuss one of the formidable open problems in the area of ffineAlgebraic Geometry, called the Epimorphism Problem. Question 1. If k[X1,...,Xn] (H) = k[n−1], then is k[X1, . . . , Xn] = k[H][n−1]? For n = 2, the answer to above question is ffirmative when k is a field of characteristic zero. This result is known as the Epimorphism Theorem proved by Abhyankar-Moh and independently by Suzuki. However, in positive characteristic there are counter examples due to Segre-Nagata. The famous Abhyankar-Sathaye conjecture asserts affirmative answer to Question 1 for n ⩾ 3 over fields of char- acteristic zero. So far we only have partial answers to this conjecture. The first affirmative result for n = 3 is due to Sathaye for linear planes over fields of char- acteristic zero. Later, Russell extended this result over fields of arbitrary character- istic. In this talk we consider the following varieties. Let m a positive integer, V an affine subvariety of Am+3 defined by a linear relation of the form xr1 1 · · · xrm m y = F (x1, . . . , xm, z, t), A the coordinate ring of V and G = Xr1 1 · · · Xrm m Y −F (X1, . . . , Xm, Z, T ). We name these varieties as “Generalised Asanuma varieties”. Earlier, Gupta had studied the case m = 1, and had obtained several necessary and sufficient con- ditions for V to be isomorphic to the affine 3-space and G to be a coordinate in k[X1, Y, Z, T ]. We study the general higher-dimensional variety V for each m ⩾ 1 and obtain analogous conditions for V to be isomorphic to Am+2 and G to be a coordinate in k[X1, . . . , Xm, Y, Z, T ], under a certain hypothesis on F . Our main theorem immediately yields a family of higher dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral do- mains of the type A under certain conditions. These results show that for each d ⩾ 3, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension d in positive characteristic. We further give complete description of two important invariants called Makar- Limanov and Derksen invariants of a certain subfamily of Generalised Asanuma varieties. In the second part of this talk we discuss about another major problem called the Cancellation Problem which investigates the following: Question 2. Let D and E be two affine domains over a field k such that D[1] =k E[1]. Does this imply D ∼=k E? The answer to Question 2 is affirmative for one dimensional affine domains. This result is due to Abhyankar, Eakin and Heinzer. However, there are coun- terexamples in dimensions greater than or equal to two. Danielewski constructed 1 a family of two dimensional pairwise non-isomorphic smooth complex varieties which are counterexamples to the Cancellation Problem. A. J. Crachiola further extended Danielewski’s examples over arbitrary characteristic. Dubouloz con- structed higher dimensional (⩾ 2) analogues of the Danielewski varieties over the field of complex numbers, which are counterexamples to this problem. Over fields of arbitrary characteristic, we establish an infinite family of a higher dimensional varieties which are pairwise non-isomorphic and are counter examples to the Can- cellation Problem. Moreover, this family accommodates the counter examples due to Dubouloz over
Algorithms and Applications in Complex Network Representation, Classification and Manipulation
No full textA complex network is a useful model for many real-world systems. Recently, much effort has been put into studying the insights of the complex network. This thesis is all about the study of complex networks. Based on the study, this thesis can be broadly divided into three parts: The first one involves analysing a complex network to find a crucial network structure called constant community by extracting and applying some features called graph representations. The second part involves the study of the quality of the graph representations on a downstream task, i.e., the node classification task. In the third part, we tried to apply the handcrafted and automatically learned graph features to some real-world scenarios, i.e., in brain networks. While detecting the constant community, we developed two strategies to construct and use the graph representations: semi-supervised and unsupervised. In the semi-supervised approach, we converted the original graph to its corresponding line graph, where a node in the line graph represents an edge in the original graph. We then applied a graph neural network (GNN) as a graph representation learning tool to classify the nodes in the line graph, which in turn was used to capture the constant communities in the original graph. In the unsupervised approach, using some hand-crafted features for each edge in the original network, we developed some novel algorithms inspired by image threshold algorithms to filter out the non-constant community edges and hence find the constant communities. In the semi-supervised approach, we noticed that when we reduced the number of training nodes, the representational capability of GNN decreased, and as a result, the classification accuracy of GNN drastically dropped. This phenomenon led us to develop input and output intervention methods to improve the accuracy of the GNN. In the input intervention, we extend the training nodes’ set using random walk and some machine learning methods to agnostically capture similar nodes from various non-contiguous sub-networks in a whole network. In the output intervention, we used random walk methods to correctly relabel the possibly misclassified nodes by the GNN as its output. The last part of the thesis deals with applications of network representation, classification, and finally manipulation in dealing with complex human brain networks. The brain regions and their interrelationships can be modelled using complex network. Utilising the complex network and its representation, in this part we contributed to neuroscience in two ways: first, we devised a methodology to diagnose a neurodevelopmental disease called Attention Deficit Hyperactivity Disorder (ADHD) using some extracted network features and applied them to various deep learning-based models. Then in the second work, we built a probabilistic model using anatomical and topological similarities to generate synthetic brain networks and track down the progression of a neurodegenerative disease called Alzheimer’s disease (AD) in human brains. The results are promising enough to establish the use of complex network analysis in computational neurolog
A Bayesian joint model for multivariate longitudinal and time-to-event data with application to ALL maintenance studies
No full textThe most common type of cancer diagnosed among children is the Acute Lymphocytic Leukemia (ALL). A study was conducted by Tata Translational Cancer Research Center (TTCRC) Kolkata, in which 236 children (diagnosed as ALL patients) were treated for the first two years (approximately) with two standard drugs (6MP and MTx) and were then followed nearly for the next 3 years. The goal is to identify the longitudinal biomarkers that are associated with time-to-relapse, and also to assess the effectiveness of the drugs. We develop a Bayesian joint model in which a linear mixed model is used to jointly model three biomarkers (i.e. white blood cell count, neutrophil count, and platelet count) and a semi-parametric proportional hazards model is used to model the time-to-relapse. Our proposed joint model can assess the effects of different covariates on the progression of the biomarkers, and the effects of the biomarkers (and the covariates) on time-to-relapse. In addition, the proposed joint model can impute the missing longitudinal biomarkers efficiently. Our analysis shows that the white blood cell (WBC) count is not associated with time-to-relapse, but the neutrophil count and the platelet count are significantly associated with it. We also infer that a lower dose of 6MP and a higher dose of MTx jointly result in a lower relapse probability in the follow-up period. Interestingly, we find that relapse probability is the lowest for the patients classified into the “high-risk” group at presentation. The effectiveness of the proposed joint model is assessed through the extensive simulation studies
A novel method addressing NGS-based mappability bias for sensitive detection of DNA alterations
No full textA turning point in cancer research is the introduction of massively parallel sequencing technology which greatly reduced the cost and time for genome sequencing. This enhanced the scope for detecting and analyzing the role of structural alterations in cancer. However, certain bias exists in NGS-based approaches, which badly affects the CNV identification process. Moreover, DNA repeats existing in CNV regions need special attention as they will degrade the performance of majority of the existing CNV detection tools, even after applying generalized bias correction method. This motivated this work, where a novel method has been designed to address the issue of DNA repeats and thereby mappability bias existing in regions of CNV. The method consists of three phases, where the first phase computes the alignment information of uniquely mapped DNA reads, considering the base quality and base mismatch parameters at nucleotide level precision. The second and the third phase use a novel approach to allocate the non-uniquely mapped reads to an optimal region of the DNA repeats based on a probabilistic membership model. The proposed method is capable of identifying CNVs present in coding, as well as non-coding region of the DNA, and is also capable of detecting CNVs existing in DNA repeat regions. The methodology achieves a sensitivity greater than 0.99 during the performed simulations, and on real data, the detected variants are validated with the database of genomic variants, where the percentage overlap is also greater than 95%, and has achieved much better breakpoint prediction, as compared with other popular bias correction CNV detection methods
A QUANTILE-REGRESSION APPROACH TO BIVARIATE LONGITUDINAL JOINT MODELING
No full textJoint modeling of longitudinal outcomes and time-to-event data has become a major research interest in the last thirty years. A joint model is useful since it helps (i) to understand the evolution of the outcome(s) of interest over time, (ii) to understand the effects of the outcomes on the time of occurrence of some event(s) of interest (e.g. death/relapse), and (iii) to study the effects of the time-varying and time-invariant predictors on the longitudinal and time-to-event process. Traditional linear mixed models are routinely used for modeling the longitudinal process. However, for non-Gaussian/skewed outcomes it is more appealing to use quantile-regression models since such models do not assume any specific probability distribution for the outcomes. In this article, we present a bivariate quantile-regression approach for jointly modeling longitudinal process and time-to-event. In a Bayesian setting we consider an Asymmetric Laplace Distribution (ALD) for modeling different quantiles of the outcomes, and a semi-parametric Proportional Hazards (PH) model for time-to-event. Model parameters are estimated using Markov chain Monte Carlo (MCMC) algorithm, and we discuss the computational complexities through several simulation studies. Our numerical studies illustrate the usefulness of our model over the other traditional models
A Review of the Rayleigh Distribution: Properties, Estimation & Application to COVID-19 Data
No full textWe study the different properties of the Rayleigh distribution. These include the descriptive properties, reliability properties and stochastic orders. Next, we consider seven different estimation methods for estimating the parameter, namely: maximum likelihood estimation, matching moments estimation, maximum product of spacing estimation, ordinary least squares estimation, Cramér-von Mises estimation, Anderson-Darling estimation and right-tail Anderson-Darling estimation. A simulation study is done to assess the performance of these methods of estimation and the results shows that all the estimators are mostly efficient and consistent. Finally, using the method of maximum likelihood estimation, we demonstrate the applicability of the Rayleigh distribution by modelling the Netherlands’s COVID-19 mortality rate data as an example
Agricultural Marketing in India: Challenges, Policies and Politics
No full textThis article is an attempt to provide a critical review of the present process of agricultural marketing in India in the wake of the recent discontent amongst the farmers that took place with the passing of the three controversial farm laws in September 2020 and giving respite to the agrarian community of the country by repealing of these new laws in November 2021. The old agricultural system of India needs to be changed. The three farm laws that were passed were of the intention to modernize the Indian agricultural market by encouraging investment and increasing competition. However, there was a country-wide protest from the farmers as they were sceptical that these laws would ultimately withdraw or reduce the security net provided by the states and put them in a vulnerable position. The present review takes a deeper dig into the present agricultural marketing situation of India in the context of the new farm laws and tries to critically evaluate the situation. JEL Classification: Q13 Q15 Q1
An Efficient Differential Grouping Algorithm for Large-Scale Global Optimization
No full textCooperative co-evolution (CC) is a practical and efficient evolutionary framework for solving large-scale global optimization problems (LSGOPs). The performance of CC depends on how variables are being grouped and can be improved through guided variable decomposition for various optimization problems. However, achieving a proper variable decomposition is computationally expensive. This article proposes an effective yet efficient differential grouping (EDG) method to reduce the associated computational cost. Our method exploits the historical interrelationship information of previous variable groups to examine interactions between the remnant variable groups. This allows us to spend less computing resources without compromising the accuracy of the final grouping result. Our proposal utilizes the covariance matrix adaptation evolution strategy (CMA-ES) algorithm, in conjunction with EDG, to solve LSGOPs. Further, to reduce time complexity and improve the stability of CMA-ES, we substitute the complex matrix decomposition step with simpler matrix operations to compute the square root of the covariance matrix. Results from our experiments and analysis indicate that EDG is a competitive method to solve LSGOPs and improve the performance of CC. The proposed schemes significantly enhance the searchability of CMA-ES compared to the other large-scale variants of CMA-ES and state-of-the-art large-scale optimizers. Moreover, our EDG could be integrated with evolutionary optimizers of different flavors like differential evolution (DE)
Application of autochthonous extremophilic Bacillus xiamenensis in remediation of groundwater- A sorption-based metal cleaning approach
No full textSurface water and groundwater used for drinking and agricultural purposes are contaminated due to anthropogenic and geogenic activities. Escalated metal concentrations, xenobiotic pollutants, competitive ions, and reusability issues are main hindrances towards decontamination of water. Moreover, the expensive purification technology brings obstacles to the underdeveloped community from availing clean water. In this context, the present work offers a sustainable cost-effective approach by providing an effective and sustainable sorption-based purification method by novel polyextremophilic bacteria Bacillus xiamenensis ISIGRM16 isolated from metal-rich industrial waste, red mud. Batch adsorption study revealed that the bacterium can remove Cd2+(\u3e99%), Ni2+(\u3e85%), and Cr6+(\u3e40%) from aqueous solution. The optimum parametric conditions for the removal of Cd2+ and Ni2+ were observed at a temperature of 30 °C and a pH of 6, while for Cr6+ removal, the optimal conditions were a temperature of 45 °C and a pH of 2. The adsorption process of Cd2+ was best explained by Freundlich isotherm (R2 ≥ 0.95), revealing multilayer adsorption. Ni2+ and Cr6+ followed the Langmuir isotherm, indicated adsorption onto the monolayer surface. The interaction mechanism was determined to be following 2nd order kinetics, with both exothermic (Cd2+, Ni2+) and endothermic (Cr6+) characteristics. The maximum adsorption capacities were found to be 31.42, 29.30, and 15.21 mg g−1 for Cd2+, Ni2+, and Cr6+ respectively. In the ternary system, the adsorption capacity followed the order Ni2+ \u3e Cd2+ \u3e Cr6+, as confirmed by both their relative adsorption capacity and from analysis visual MINTEQ. Microscopic and spectroscopic analyses revealed, altered cell morphology, metal deposition on the bacterial cell surface, and the involvement of hydroxyl, carboxyl, and amide groups in the elimination of Cd2+, Ni2+, and Cr6+. Further the sequential adsorption-desorption study confirmed a significantly preserved removal efficacy (p \u3c 0.05), indicating the advantageous use of the bacterium as a biosorbent
Calophyllum L.: An important tropical element in the monsoon-influenced ancient Siwalik Forest of eastern Himalaya
No full textCalophyllum vegetative (leaves and woods) fossil remains have been widely reported from the Cenozoic sedimentary strata across the Northern and Southern Hemispheres. However, until now, no reproductive megafossil remains have been discovered. Here, we report and describe Calophyllum fossil fruits from the Siwalik (middle Miocene–Pliocene) sediments of Darjeeling foothills, eastern Himalaya for the first time. In addition, we also provide fossil materials of isolated leaves in appreciable numbers recovered from the same stratigraphic level. Our Siwalik fossils and extant members of Calophyllum are closely related morphologically by stalked, ball-shaped (spherical to ovoid) drupe with a thin, dark brown, smooth surfaced outer layer and a hard endocarp; and symmetrical, oblong-elliptic shaped entire margined lamina with numerous close parallel secondary veins, and obscure tertiary veins. Based upon morphology and epidermal anatomy, the fossil fruits are recognized as a new fossil-species Calophyllum ramthiene sp. nov. The present Calophyllum leaf and fruit fossils, along with previous reports suggest that Calophyllum was an important forest element throughout the eastern Himalaya during the period of Siwalik sedimentation (Mio-Pleistocene time). We briefly discuss the fossil history of Calophyllum and palaeobiogeography in India and palaeoclimatic implications regarding the distribution and habitat of fossil and modern members of Calophyllum